3frac (problem 3.3.3)

Average Error: 9.7 → 0.3

Time: 17.6s

Precision: binary64

Cost: 27076

Math

\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -47:\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{9}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{7}}\right)\right)\right)\\ \mathbf{elif}\;x \leq 12200:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

↓

(FPCore (x)
 :precision binary64
 (if (<= x -47.0)
   (*
    2.0
    (+
     (/ 1.0 (pow x 3.0))
     (+ (/ 1.0 (pow x 9.0)) (+ (/ 1.0 (pow x 5.0)) (/ 1.0 (pow x 7.0))))))
   (if (<= x 12200.0)
     (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0)))
     (/ 2.0 (pow x 3.0)))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

↓

double code(double x) {
	double tmp;
	if (x <= -47.0) {
		tmp = 2.0 * ((1.0 / pow(x, 3.0)) + ((1.0 / pow(x, 9.0)) + ((1.0 / pow(x, 5.0)) + (1.0 / pow(x, 7.0)))));
	} else if (x <= 12200.0) {
		tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
	} else {
		tmp = 2.0 / pow(x, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-47.0d0)) then
        tmp = 2.0d0 * ((1.0d0 / (x ** 3.0d0)) + ((1.0d0 / (x ** 9.0d0)) + ((1.0d0 / (x ** 5.0d0)) + (1.0d0 / (x ** 7.0d0)))))
    else if (x <= 12200.0d0) then
        tmp = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
    else
        tmp = 2.0d0 / (x ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

↓

public static double code(double x) {
	double tmp;
	if (x <= -47.0) {
		tmp = 2.0 * ((1.0 / Math.pow(x, 3.0)) + ((1.0 / Math.pow(x, 9.0)) + ((1.0 / Math.pow(x, 5.0)) + (1.0 / Math.pow(x, 7.0)))));
	} else if (x <= 12200.0) {
		tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
	} else {
		tmp = 2.0 / Math.pow(x, 3.0);
	}
	return tmp;
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

↓

def code(x):
	tmp = 0
	if x <= -47.0:
		tmp = 2.0 * ((1.0 / math.pow(x, 3.0)) + ((1.0 / math.pow(x, 9.0)) + ((1.0 / math.pow(x, 5.0)) + (1.0 / math.pow(x, 7.0)))))
	elif x <= 12200.0:
		tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))
	else:
		tmp = 2.0 / math.pow(x, 3.0)
	return tmp

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

↓

function code(x)
	tmp = 0.0
	if (x <= -47.0)
		tmp = Float64(2.0 * Float64(Float64(1.0 / (x ^ 3.0)) + Float64(Float64(1.0 / (x ^ 9.0)) + Float64(Float64(1.0 / (x ^ 5.0)) + Float64(1.0 / (x ^ 7.0))))));
	elseif (x <= 12200.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)));
	else
		tmp = Float64(2.0 / (x ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -47.0)
		tmp = 2.0 * ((1.0 / (x ^ 3.0)) + ((1.0 / (x ^ 9.0)) + ((1.0 / (x ^ 5.0)) + (1.0 / (x ^ 7.0)))));
	elseif (x <= 12200.0)
		tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
	else
		tmp = 2.0 / (x ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := If[LessEqual[x, -47.0], N[(2.0 * N[(N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[Power[x, 9.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 12200.0], N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]]]

Target

Original	9.7
Target	0.3
Herbie	0.3

\[\frac{2}{x \cdot \left(x \cdot x - 1\right)} \]

Derivation

1. Split input into 3 regimes

2. if x < -47

   1. Initial program 18.5

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   2. Applied egg-rr 22.2

      \[\leadsto \color{blue}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \left(\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \frac{1}{\frac{1}{1 + x} - \frac{2}{x}}\right)} + \frac{1}{x - 1} \]

   3. Taylor expanded in x around inf 0.5

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{9}}\right)\right)} \]

   4. Simplified 0.5

      \[\leadsto \color{blue}{2 \cdot \left(\frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{9}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{7}}\right)\right)\right)} \]
      Proof

      [Start] 0.5	\[ 2 \cdot \frac{1}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{9}}\right)\right) \]
      rational_best-simplify-43 [=>] 0.5	\[ \color{blue}{\left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{9}}\right) + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{5}}\right)} \]
      rational_best-simplify-47 [=>] 0.5	\[ \color{blue}{2 \cdot \left(\frac{1}{{x}^{9}} + \frac{1}{{x}^{3}}\right)} + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{5}}\right) \]
      rational_best-simplify-47 [=>] 0.5	\[ 2 \cdot \left(\frac{1}{{x}^{9}} + \frac{1}{{x}^{3}}\right) + \color{blue}{2 \cdot \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{7}}\right)} \]
      rational_best-simplify-47 [=>] 0.5	\[ \color{blue}{2 \cdot \left(\left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{7}}\right) + \left(\frac{1}{{x}^{9}} + \frac{1}{{x}^{3}}\right)\right)} \]
      rational_best-simplify-43 [=>] 0.5	\[ 2 \cdot \color{blue}{\left(\frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{9}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{7}}\right)\right)\right)} \]

   if -47 < x < 12200

   1. Initial program 0.1

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   if 12200 < x

   1. Initial program 20.3

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   2. Taylor expanded in x around inf 0.6

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -47:\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{9}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{7}}\right)\right)\right)\\ \mathbf{elif}\;x \leq 12200:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	20356

\[\begin{array}{l} \mathbf{if}\;x \leq -126:\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{5}} + \left(\frac{1}{{x}^{3}} + \frac{1}{{x}^{7}}\right)\right)\\ \mathbf{elif}\;x \leq 12200:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \end{array} \]

Alternative 2
Error	0.9
Cost	15560

\[\begin{array}{l} t_0 := \frac{1}{x + 1}\\ t_1 := \frac{1}{x - 1}\\ t_2 := \left(t_0 - \frac{2}{x}\right) + t_1\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\left(\frac{2}{x} + \left(t_0 + -2 \cdot \frac{2}{x}\right)\right) + t_1\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 3
Error	0.9
Cost	8712

\[\begin{array}{l} t_0 := \frac{1}{x + 1}\\ t_1 := \frac{1}{x - 1}\\ t_2 := \left(t_0 - \frac{2}{x}\right) + t_1\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\left(\frac{2}{x} + \left(t_0 + -2 \cdot \frac{2}{x}\right)\right) + t_1\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 4
Error	9.7
Cost	960

\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

Alternative 5
Error	10.5
Cost	448

\[\left(1 - \frac{2}{x}\right) + -1 \]

Alternative 6
Error	30.8
Cost	192

\[\frac{-2}{x} \]

Alternative 7
Error	61.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023094 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))